Ternary function in axiomatization of category theory

Question

I am interested in axiomatizing category theory (for example, the category of categories or category of sets just like Lawvere did). I went through some sources and everywhere I looked I saw that the composition was axiomatized instead of $\circ(u,v)$ which would be binary function but using ternary function $\Gamma(u,v,w)$ which is interpreted in metamathematics as "$u \circ v = w$". Why is that so? I don't see why we choose this because I feel that $\circ$ is a more natural choice and equality is already assumed while axiomatizing category theory.

Answer 1

In traditional first-order logic, functions must be total. So, if $\circ$ were a binary operation, $\circ(u,v)$ would need to be defined for any two morphisms $u$ and $v$. This is obviously not desirable, since composition should only be defined when the domain of one morphism matches the codomain of the other. So, since it is not a total function, you must encode $\circ$ instead as a relation.

See Model theory: What is the signature of `Category theory` for some related discussion.